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Astronomy

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Astronomical Distance Units .
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Celestial Coordinates .
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Celestial Navigation .
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Location of North and South Celestial Poles .

Chemistry

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Avogadro's Number
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Balancing Chemical Equations
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Stochiometry
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The Periodic Table .

Classical Physics

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Archimedes Principle
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Bernoulli Principle
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Blackbody (Cavity) Radiation and Planck's Hypothesis
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Center of Mass Frame
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Comparison Between Gravitation and Electrostatics
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Compton Effect .
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Coriolis Effect
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Cyclotron Resonance
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Dispersion
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Doppler Effect
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Double Slit Experiment
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Elastic and Inelastic Collisions .
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Electric Fields
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Error Analysis
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Fick's Law
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Fluid Pressure
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Gauss's Law of Universal Gravity .
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Gravity - Force and Acceleration
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Hooke's law
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Ideal and Non-Ideal Gas Laws (van der Waal)
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Impulse Force
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Inclined Plane
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Inertia
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Kepler's Laws
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Kinematics
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Kinetic Theory of Gases .
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Kirchoff's Laws
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Laplace's and Poisson's Equations
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Lorentz Force Law
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Maxwell's Equations
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Moments and Torque
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Nuclear Spin
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One Dimensional Wave Equation .
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Pascal's Principle
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Phase and Group Velocity
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Planck Radiation Law .
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Poiseuille's Law
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Radioactive Decay
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Refractive Index
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Rotational Dynamics
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Simple Harmonic Motion
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Specific Heat, Latent Heat and Calorimetry
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Stefan-Boltzmann Law
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The Gas Laws
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The Laws of Thermodynamics
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The Zeeman Effect .
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Wien's Displacement Law
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Young's Modulus

Climate Change

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Keeling Curve .

Cosmology

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Baryogenesis
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Cosmic Background Radiation and Decoupling
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CPT Symmetries
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Dark Matter
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Friedmann-Robertson-Walker Equations
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Geometries of the Universe
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Hubble's Law
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Inflation Theory
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Introduction to Black Holes .
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Olbers' Paradox
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Penrose Diagrams
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Planck Units
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Stephen Hawking's Last Paper .
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Stephen Hawking's PhD Thesis .
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The Big Bang Model

Finance and Accounting

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Amortization
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Annuities
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Brownian Model of Financial Markets
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Capital Structure
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Dividend Discount Formula
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Lecture Notes on International Financial Management
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NPV and IRR
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Periodically and Continuously Compounded Interest
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Repurchase versus Dividend Analysis

Game Theory

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The Truel .

General Relativity

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Accelerated Reference Frames - Rindler Coordinates
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Catalog of Spacetimes .
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Curvature and Parallel Transport
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Dirac Equation in Curved Spacetime
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Einstein's Field Equations
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Geodesics
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Gravitational Time Dilation
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Gravitational Waves
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One-forms
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Quantum Gravity
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Relativistic, Cosmological and Gravitational Redshift
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Ricci Decomposition
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Ricci Flow
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Stress-Energy Tensor
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Stress-Energy-Momentum Tensor
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Tensors
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The Area Metric
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The Equivalence Principal
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The Essential Mathematics of General Relativity
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The Induced Metric
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The Metric Tensor
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Vierbein (Frame) Fields
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World Lines Refresher

Lagrangian and Hamiltonian Mechanics

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Classical Field Theory .
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Euler-Lagrange Equation
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Ex: Newtonian, Lagrangian and Hamiltonian Mechanics
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Hamiltonian Formulation .
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Liouville's Theorem
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Symmetry and Conservation Laws - Noether's Theorem

Macroeconomics

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Lecture Notes on International Economics
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Lecture Notes on Macroeconomics
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Macroeconomic Policy

Mathematics

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Amplitude, Period and Phase
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Arithmetic and Geometric Sequences and Series
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Asymptotes
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Augmented Matrices and Cramer's Rule
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Basic Group Theory
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Basic Representation Theory
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Binomial Theorem (Pascal's Triangle)
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Building Groups From Other Groups
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Completing the Square
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Complex Numbers
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Composite Functions
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Conformal Transformations .
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Conjugate Pair Theorem
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Contravariant and Covariant Components of a Vector
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Derivatives of Inverse Functions
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Double Angle Formulas
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Eigenvectors and Eigenvalues
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Euler Formula for Polyhedrons
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Factoring of a3 +/- b3
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Fourier Series and Transforms .
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Fractals
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Gauss's Divergence Theorem
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Grassmann and Clifford Algebras .
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Heron's Formula
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Index Notation (Tensors and Matrices)
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Inequalities
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Integration By Parts
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Introduction to Conformal Field Theory .
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Inverse of a Function
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Law of Sines and Cosines
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Line Integrals, ∮
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Logarithms and Logarithmic Equations
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Matrices and Determinants
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Matrix Exponential
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Mean Value and Rolle's Theorem
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Modulus Equations
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Orthogonal Curvilinear Coordinates .
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Parabolas, Ellipses and Hyperbolas
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Piecewise Functions
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Polar Coordinates
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Polynomial Division
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Quaternions 1
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Quaternions 2
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Regular Polygons
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Related Rates
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Sets, Groups, Modules, Rings and Vector Spaces
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Similar Matrices and Diagonalization .
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Spherical Trigonometry
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Stirling's Approximation
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Sum and Differences of Squares and Cubes
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Symbolic Logic
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Symmetric Groups
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Tangent and Normal Line
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Taylor and Maclaurin Series .
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The Essential Mathematics of Lie Groups
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The Integers Modulo n Under + and x
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The Limit Definition of the Exponential Function
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Tic-Tac-Toe Factoring
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Trapezoidal Rule
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Unit Vectors
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Vector Calculus
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Volume Integrals

Microeconomics

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Marginal Revenue and Cost

Particle Physics

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Feynman Diagrams and Loops
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Field Dimensions
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Helicity and Chirality
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Klein-Gordon and Dirac Equations
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Regularization and Renormalization
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Scattering - Mandelstam Variables
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Spin 1 Eigenvectors .
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The Vacuum Catastrophe

Probability and Statistics

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Box and Whisker Plots
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Categorical Data - Crosstabs
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Chebyshev's Theorem
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Chi Squared Goodness of Fit
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Conditional Probability
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Confidence Intervals
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Data Types
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Expected Value
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Factor Analysis
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Hypothesis Testing
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Linear Regression
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Monte Carlo Methods
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Non Parametric Tests
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One-Way ANOVA
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Pearson Correlation
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Permutations and Combinations
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Pooled Variance and Standard Error
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Probability Distributions
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Probability Rules
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Sample Size Determination
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Sampling Distributions
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Set Theory - Venn Diagrams
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Stacked and Unstacked Data
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Stem Plots, Histograms and Ogives
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Survey Data - Likert Item and Scale
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Tukey's Test
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Two-Way ANOVA

Programming and Computer Science

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Hashing
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How this site works ...
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More Programming Topics
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MVC Architecture
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Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
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Public Key Encryption

Quantum Computing

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The Qubit .

Quantum Field Theory

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Creation and Annihilation Operators
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Field Operators for Bosons and Fermions
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Lagrangians in Quantum Field Theory
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Path Integral Formulation
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Relativistic Quantum Field Theory

Quantum Mechanics

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Basic Relationships
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Bell's Theorem
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Bohr Atom
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Clebsch-Gordan Coefficients .
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Commutators
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Dyson Series
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Electron Orbital Angular Momentum and Spin
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Entangled States
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Heisenberg Uncertainty Principle
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Ladder Operators .
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Multi Electron Wavefunctions
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Pauli Exclusion Principle
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Pauli Spin Matrices
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Photoelectric Effect
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Position and Momentum States
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Probability Current
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Schrodinger Equation for Hydrogen Atom
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Schrodinger Wave Equation
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Schrodinger Wave Equation (continued)
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Spin 1/2 Eigenvectors
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The Differential Operator
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The Essential Mathematics of Quantum Mechanics
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The Observer Effect
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The Quantum Harmonic Oscillator .
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The Schrodinger, Heisenberg and Dirac Pictures
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The WKB Approximation
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Time Dependent Perturbation Theory
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Time Evolution and Symmetry Operations
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Time Independent Perturbation Theory
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Wavepackets

Semiconductor Reliability

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The Weibull Distribution

Solid State Electronics

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Band Theory of Solids .
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Fermi-Dirac Statistics .
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Intrinsic and Extrinsic Semiconductors
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The MOSFET
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The P-N Junction

Special Relativity

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4-vectors .
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Electromagnetic 4 - Potential
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Energy and Momentum, E = mc2
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Lorentz Invariance
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Lorentz Transform
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Lorentz Transformation of the EM Field
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Newton versus Einstein
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Spinors - Part 1 .
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Spinors - Part 2 .
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The Lorentz Group
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Velocity Addition

Statistical Mechanics

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Black Body Radiation
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Entropy and the Partition Function
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The Harmonic Oscillator
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The Ideal Gas

String Theory

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Bosonic Strings
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Extra Dimensions
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Introduction to String Theory
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Kaluza-Klein Compactification of Closed Strings
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Strings in Curved Spacetime
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Toroidal Compactification

Superconductivity

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BCS Theory
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Introduction to Superconductors
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Superconductivity (Lectures 1 - 10)
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Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

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Chiral Superfields
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Generators of a Supergroup
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Grassmann Numbers
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Introduction to Supersymmetry
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The Gauge Hierarchy Problem

The Standard Model

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Electroweak Unification (Glashow-Weinberg-Salam)
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Gauge Theories (Yang-Mills)
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Gravitational Force and the Planck Scale
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Introduction to the Standard Model
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Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
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Quantum Flavordynamics and Quantum Chromodynamics
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Special Unitary Groups and the Standard Model - Part 1 .
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Special Unitary Groups and the Standard Model - Part 2
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Special Unitary Groups and the Standard Model - Part 3 .
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Standard Model Lagrangian
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The Higgs Mechanism
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The Nature of the Weak Interaction

Topology

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Units, Constants and Useful Formulas

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Constants
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Formulas
Last modified: January 26, 2018

Complex Scalar Fields --------------------- Think of 2 real scalar fields φ1 and φ2 and combine them together in the following manner. φ = φ1 + iφ2 and φ* = φ1 - iφ2 We can write that ∂φ/∂xμ = ∂φ1/∂xμ + i∂φ2/∂xμ . L = φ12/2 - (∂xφ1/2)2 - (∂yφ1/2)2 - (∂zφ1/2)2 - m2φ12 . + φ22/2 - (∂xφ2/2)2 - (∂yφ2/2)2 - (∂zφ2/2)2 - m2φ22 This can be simplified to: .. = φφ*/2 - ∂xφ*xφ - ∂yφ*yφ - ∂zφ*xφ - m2φ*φ/2 It is conventional to drop the 1/2. Therefore, in abbreviated form, we can write: L = ∂μφ∂μφ* - m2φ*φ The quantity m2φ*φ is referred to as the 'mass term'. Now apply a phase transformation, exp(iθ), to φ. This can be considered as a 'rigid' or global transformation that changes all points in the field by the same amount. It is easy to see that the Lagrangian is invariant under this transformation (φ' = φexp(iθ) and φ'* = φ*exp(-iθ)). This implies there is a symmetry and a conserved quantity. Now e = 1 + iε ... Taylor series Thus, φ -> φ + iεφ -> φ + εfφ where fφ = iφ φ* -> φ* - iεφ* -> φ* - εfφ* where fφ* = -iφ* Look for a conserved quantity using Noether's Theorem. . . πφ = ∂L/∂φ = φ* . . πφ* = ∂L/∂φ* = φ Conserved quantity = ∫[πφfφ + πφ*fφ*]d3x . . = i∫[φ*φ - φφ*]d3x The quantity in [] is the charge density of the field, ρ. Another way to write this is: . ρ = Im(φφ*) ∴ Q = ∫ρd3x We can also write a 4-vector for current, jμ, as: jμ = Im(φ*μφ) The word 'charge' can have several different meanings. It can be the electric charge but also the weak hypercharge. The take away from all of this is that the rotational symmetry of a field is associated with charge. THUS, THE U(1) SYMMETRY IS ASSOCIATED WITH CHARGE. Notes: - Every field has a field quanta. E-M field => photon. - Every particle is a quanta of a field. - CHARGED PARTICLES ARE ALWAYS DESCRIBED BY COMPLEX FIELDS. - Don't confuse ρ with the e-m field. They are different things. - Need to couple ρ with e-m field in a Lagrangian to see how they interact. Gauge Theory ------------ A symmetry is a transformation that preserves relevant physical properties such as the equations of motion. It can be described as a transformation that leaves the Lagrangian invariant. A GLOBAL symmetry is a symmetry transformation that is executed by the same 'amount' at every point. A LOCAL symmetry has a transformation that depends on space-time. In the case of rotation, the angle of the rotation can change for different locations and times A gauge symmetry is a local symmetry that relates states that are physically the same. The states it connects are then only different in their mathematical structure. The presence of a gauge symmetry can be viewed as a redundancy in description. In the words there is more description and more mathematical structure than is strictly necessary to describe the physical state. Gauge transformations represent a stronger symmetry than the global symmetry associated with a rigid transformation where θ is not a function of position. We will see how the use of a gauge transformation can make an otherwise invariant Lagrangian invariant. Consider a complex field φ = φR + iφI and φ* = φR - iφI and rotate by a phase θ(x) to get φ' as follows: φ' = φeiθ(x) and φ'* = φ*e-iθ(x) This is a local U(1) symmetry. Recall from the previous section on complex fields that: L = ∂μφ'∂μφ'* - m2φ'φ'* After some math the Lagrangian density becomes (dropping the factor of 1/2 to simplify matters): (∂μφ - i∂μθφ)(∂μφ* + i∂μθφ*) - m2φφ* The m2φφ'* term is invariant but the rest is not. In order to make the Lagrangian invariant under the local U(1) symmetry, we need to add the concept of the GAUGE COVARIANT DERIVATIVE, Dμ. For the electromagnetic field Dμ is defined as: Dμφ = ∂μφ + iqAμφ and Dμφ* = ∂μφ* - iqAμφ* where Aμ is the GAUGE FIELD written as a 4-vector = (A0,A1,A2,A3). In the case of the electromagnetic field A0, is the electrostatic potential and A1..3 is the magnetic vector potential E = -∇A0 - ∂A1..3/∂τ and B = ∇xA1..3. Note: Aμ itself is not itself gauge invariant. q is the charge (coupling constant). The electric charge is just a measure of the strength of a particle's interaction with the electromagnetic field. Saying that a particle has a given charge is the same as saying it interacts with the electromagnetic field with a certain strength. In order for the Lagrangian density to be invariant it is necessary that A transform as: Aμ' = Aμ - (1/q)∂μθ Proof: Dμφ' = ∂μφ' + iqAμ'φ' = (∂μφ + iφ∂θ/∂x)eiθ(x) + iq(Aμφ - (1/q)φ∂θ/∂x)eiθ(x) = (∂μφ + iqAμφ)eiθ(x) = eiθ(x)Dμφ Similarly, Dμφ'* = e-iθ(x)Dμφ* Therefore, Dμφ'Dμφ'* = DμφDμφ* and the Lagrangian is gauge invariant. We can rewrite the Lagrangian as: L = DμφDμφ* - m2φφ* = (∂μφ + iqAμφ)(∂μφ* - iqAμφ*)eiθ(x)e-iθ(x) - m2φφ* = ∂μφ∂μφ* + q2AμAμφφ* + iqAμ[φ∂μφ* - φ*μφ] - m2φφ* This Langrangian is invariant under the local U(1) symmetry. The [] term represents a current and Aμ[] represents the interaction between this current with the electromagnetic field. Thus, we see that our theory which is invariant under local gauge transformations is promoted to an interacting theory. Invariance of Aμ by Itself -------------------------- We have looked at how the introduction of the gauge field makes φ invariant under a local transformation. However, the gauge field itself must also have its own gauge invariant dynamic term in the Lagrangian. Consider the construction, Fμν, defined as: Fμν = ∂μAν - ∂νAμ Intuitively, this is a good choice since it contains the first derivatives, ∂μ, of the field that is characterisitic of all Lagrangians. If we plug in Aμ' = Aμ - (1/q)∂μθ into the above we get: Fμν = ∂μAν + ∂μνθ - (∂νAμ + ∂νμθ)   = ∂μAν - ∂νAμ Thus, under the gauge transformation, Fμν, remains unchanged. To make this both gauge and Lorentz invariant and also match the quadratic nature of the other terms in the Lagrangian, it is necessary to form the product: FμνFμν We can now rewite our a Lagrangian for the gauge field as: L = -FμνFμν The is the Lagrangian that describes that electromagnetic field in the absence of any charges (complex fields are charged fields so in the absence of charge φ = 0). The Higgs Mechanism -------------------- In the previous section we have talked about the role of gauge fields in constucting invariant Lagrangians. The quanta associated with these fields are called GAUGE BOSONS. By analogy with the m2φφ* term associated with the complex field, we might expect that mass for the gauge bosons might be associated with a m2AμAμ term. However, such terms are not invariant and therefore break the local symmetry. The invariance can be seen since m2(Aμ + ∂μθ)(Aμ + ∂μθ) is clearly not equal to m2AμAμ. The q2AμAμφφ* term from before represents an interaction term and IS invariant because Aμ and Aμ have already been transformed during the derivation. The consequence of this invariance is that, in general, we can make the statement that gauge symmetry prohibits gauge bosons from having mass. The standard model is a gauge theory and particles do have mass. The implication of this is that for mass to appear in the standard model, the local symmetry has to be broken. This is referred to as SPONTANEOUS SYMMETRY BREAKING. Bosonic Mass ------------ Nearly all of the mass of protons and neutrons is from the kinetic energy of their constituent quarks and gluons. This kinetic energy can only exist if the quarks themselves have a rest mass. It is the Higgs field that is responsible for giving mass to these constituent particles and also to the electrons. The Higgs field is also responsible for giving mass to the REAL W+, W- and Z0 gauge bosons that can be produced in collisions if sufficient energy is available. The potential energy, V(φ), of the Higgs field has the shape of a 'Mexican hat'. For ordinary fields the lowest energy is the point corresponding to φ = 0. For the Higgs, however, the location of the lowest energy is not at φ = 0 but at the point φ = f. Any system, including a universe, will tumble into its lowest energy state, like a ball bouncing down to the bottom of a valley. The universe, like a ball, comes to rest somewhere on this circular trench, which corresponds to a nonzero value of the field. That is, in its natural, lowest energy state, the universe is permeated throughout by a nonzero Higgs field. The Higgs field is a complex field represented by: φ = φR + iφI and φ* = φR - iφI The equation representing the potential of the Higgs field is of the form: V(φ) = -m2φ2 + λφ4 where λ is a coupling constant. Note: +m2φ2 is a parabaloid centered at 0. For the minima: ∴ dV(φ)/dφ = -2m2φ + 4λφ3 = 0 so that, φ = f = m/√2λ f is referred to as the VACUUM EXPECTATION VALUE. The VEV for the Higgs field is 246 GeV. The Higgs potential can also be written as: V(φ) = -m2φφ*+ λ(φφ*)2 In this system, polar coordinates are much easier to work with than cartesian coordinates. We can write: φ = ρe and φ* = ρe-iα Where ρ represents the radial coordinate and α represents the angular coordinate. Abelian U(1) Gauge Theory ------------------------- The particle at point A is symmetrical about rotation of the complex plane. Thus, the Higgs field has U(1) symmetry. When this symmetry is spontaneously broken, the particle 'descends' into the trough (point B) where the energy is at a minimum. At this point the particle experiences a flat potential as it moves along a circular path as shown. A movement in this direction does not face any resistance since the energy in the adjacent state is the same. A particle moving in this angular fashion therefore has zero mass. This is the GOLDSTONE boson*. In contrast, oscillations back and forth along the radial (red) direction are not massless. These are the HIGGS boson excitations. Mathematically, we can get some insight into the Higgs mechanism as follows: *Goldstone's Theorem: For each broken generator of the original symmetry group one massless spin-zero particle will appear. We can write the Lagrangian for a spin 0 field (the Higgs boson is spin 0 particle) as: L = ∂μφ∂μφ* - V(φ) = (∂μφR)2 + (∂μφI)2 - V(φ) In terms of ρ and α this is equivalent to: L = (∂μρ)2 + ρ2(∂μα)2 - V(ρ) Now ρ = f and is fixed and V(ρ) -> 0. Therefore, the Lagrangian can be written as: L = 0 + f2(∂μα)2 - 0 = f2(∂μα)2 = (∂μfα)2 This is characteristic of a field with quanta of zero mass. This is the GOLDSTONE FIELD. ρ represents the distance to the bottom of the trough, f, plus a small displacement, H. Thus we can write: ρ = f + H The amount of energy required to excite the Higgs is quite large. If we consider a low energy where the oscillations are small we can write: φ = fe Under a local U(1) transformation the field transforms as: φ -> φexp(iθ(x)) However, this is not symmetric since ∂μφ∂μφ* ≠ ∂μφ'∂μφ'*. In order to make it so we need to introduce the GAUGE FIELD, Aμ where Aμ transforms as Aμ -> Aμ + ∂μθ and use the covariant derivative. This results in: Dμφ = ∂μφ + iqAμφ = i(∂μα + qAμ)fe Likewise, Dμφ* = -i(∂μα + qAμ)fe-iα Therefore, DμφDμφ* = (qAμ + ∂μα)2f2     = q2f2AμAμ + 2qf2Aμμα + f2(∂μα)2 If we now make the UNITARY gauge transformation Aμ -> Aμ - (1/q)∂μα we get: DμφDμφ* = q2f2[Aμ - (1/q)∂μα][Aμ - (1/q)∂μα] + 2qf2[Aμ - (1/q)∂μα]∂μα + f2(∂μα)2     = q2f2AμAμ - 2qf2Aμμα + f2(∂μα)2 + 2qf2Aμμα - 2f2(∂μα)2 + f2(∂μα)2     = q2f2AμAμ     ≡ mγ2AμAμ So,'mathemagically' the Goldstone boson disappears. This explains why it is never seen in nature. In the process of disappearing, the Goldstone boson has given rise to a mass m2A2 equivalent to the mass of a photon. It is instructive to count the degrees of freedom before and after SSB has occurred. We started out with a massless photon with 2 degrees of freedom and a complex scalar field with 2 degrees of freedom for a total number of 4 degrees of freedom. After the SSB we have one massive photon with 3 degrees of freedom and a real scalar field with 1 degree of freedom, again for a total of 4 degrees of freedom. So this is a little confusing since we know the 'everyday' photon is massless. We must conclude that spontaneous symmetry breaking does not occur for the photon, or does it?. There is a special case where spontaneous symmetry breaking does occur and that is within a superconductor. In the theory of superconductivity the photon does in fact have a mass. So far we have assumed that the H in f + H is small and oscillations of the field can be ignored. However, if the system is given enough energy, the field will start oscillating in the radial direction along ρ (the red curve). These oscillations represent the HIGGS BOSON. As before, we can choose a direction of fluctuation so that the vacuum Higgs field is: φ = (f + H)e The Lagrangian now looks like: L = Dμ{(f + H)e}Dμ{(f + H)*e-iα} - m2(f + H)2 - λ(f + H)4 - FμνFμν = (∂μ + iqAμ)(fe + He)(∂μ - iqAμ)(fe-iα + He-iα) - m2(f + H)2 - λ(f + H)4 - FμνFμν After some math The Lagrangian becomes: L = q2f2AμAμ + 2qf2Aμμα + f2(∂μα)2 + ∂μH∂μH - 4λf2H2 - FμνFμν + cubic and quartic terms This Lagrangian now describes a theory with a photon of mass mγ = qf, a Higgs boson, H, with mH = 2√(λ)f and a massless Goldstone boson, α. Again, we can make the UNITARY gauge transformation Aμ -> Aμ - (1/q)∂μα to get: L = q2f2AμAμ + ∂μH∂μH - 4λf2H2 - FμνFμν + cubic and quartic terms Now there are no massless particles in this theory, and the α field has completely disappeared from the Lagrangian. Thus, when the Higgs field is excited, the Higgs boson also get its mass as a consequence of spontaneous symmetry breaking. Note: It is inportant to distinguish between the Higgs field and the Higgs boson. The Higgs field is the entity that gives all other particles a mass. The Higgs boson is a particle (quanta of the field) that gets its mass like all other particles. Once the Higgs particle has been created, it will eventually decay. Though the Higgs particle interacts with all massive particles it prefers to interact with the heaviest elementary particles. The Higgs particle is considered to be a carrier of a force. It is a boson, like the other force-transferring particles: photons, gluons and electroweak bosons. It has 0 spin, carries no charge and has a mass of about 126 GeV. Non-Abelian SU(2)⊗U(1) Gauge Theory (Yang-Mills) ------------------------------------------------ In the interest of simplicity, the above arguments have only considered the Abelian U(1) gauge symmetry breaking and are by no means rigorous. The discussion has been focussed around the photon and the mechanism for gaining mass. The sole purpose has been only to provide a basic idea of the fundamental concepts and in this respect the Abelian U(1) model is sufficient. For the SU(2) symmetry that occurs in the electroweak theory of the Standard Model, the elements in the Lagrangian are a bit different. The gauge symmetric theory for SU(2) was developed by YANG and MILLS. Under a local SU(2) transformation the field transforms as: φ -> φexp(iθi(x)σi/2) where σi are the Pauli matrices - the generators of SU(2). Note: These are NOT the spin angular momentum generators. They are, instead, asscociated with the weak isospin. The physical interpretation is completely different, but the algebra is identical to that of angular momentum. θi are the 3 group parameters. The covariant derivative for a SU(2)⊗U(1) gauge invariant Lagrangian becomes: Dμ = ∂μ + igWμiσi/2 + ig'YWBμ Where g and g' are coupling constants. The Wμi fields are associated with the weak isospin, the generator of SU(2). Since there are 3 generators, there are 3 gauge fields (with massless bosons) designated by Wμ0, Wμ1 and Wμ2. Likewise the Bμ field is associated with the weak hypercharge, YW, the generator of U(1). Note that this U(1) is NOT the same as the U(1) in the Abelian case discussed above for the photon. In that case, U(1) is associated with the conventional charge, q. By analogy with photons, particles with weak hypercharge interact by the exchange of B bosons - the massless quanta of the B gauge field. Due to the non-commutativity of the generators of this symmetry ([σ12] = iσ3 etc.), the gauge tensor for the W fields is: Gμνi = ∂μWνi - ∂νWμi + fijkWμjWνk where fijk is the structure constant of the particular gauge group. The basic form of the Langrangian that encapsulates SU(2)⊗U(1) looks like (omitting constants): L = DμφDμφ - V(φ) - FμνFμν - GμνiGμνi where Fμν has been redefined as: Fμν = ∂μBν - ∂νBμ In the Abelian case the Higgs field was considered to be a complex scalar field with 2 components. In the non-Abelian case the Higgs field is a complex scalar field of the group SU(2) with 4 components - 2 neutral (0) and 2 charged (Q) each with 1 degree of freedom for a total of 4. Neglecting normalization factors, the field is a weak isospin doublet written as: - - - - - - φ = | φ+ | = | φ1 + iφ2 | ≡ | H+ + iH- | with I3 = 1/2 and YW = 1/2 | φ0 | | φ3 + iφ4 | | H  + iH 0| with I3 = -1/2 and YW = 1/2 - - - - - - and its Hermitian conjugate, - - φ = | φ- φ0* | - - Under U(1) rotations, these doublets are multiplied by a phase exp(iφYW) Gauge symmetry is spontaneously broken down to the U(1) symmetry of electromagnetism. This can be written as: SU(2)L⊗U(1)Y -> U(1)EM Where L denotes left-handed chirality, YW denotes the weak hypercharge and EM denotes the electromagnetic field. As in the Abelian case, for low energies (H very small) we can write: DμφDμφ = (∂μφ + igWμiσiφ/2 + ig'YWBμφ)(∂μφ - igWμiσiφ/2 - ig'YWBμφ) After some math and, for the purpose of illustration, neglecting all constants, we get: DμφDμφ = f2[g2{(Wμ1)2 + (Wμ2)2} + (gWμ0 - g'Bμ)2] The g2 term can be written as WμWμ where Wμ and Wμ are LADDER operators defined as: Wμ = Wμ1 + iWμ2 = W- Wμ = Wμ1 - iWμ2 = W+ Ladder operators were introduced in the context of angular momentum and rotation so it is also appropriate to use them in the context of rotations associated with SU(2). Thus, we can write: f2g2{(Wμ1)2 + (Wμ2)2} = f2g2WμWμ and identify the mass mW = gf The 2 remaining neutral Z and γ bosons are defined in terms of the WEINBERG ANGLE, θW, as follows: Zμ0 = Wμ0cosθW - BμsinθW γμ = Wμ0sinθW + BμcosθW θW is found experimentally and is the angle by which SSB rotates the original Wμ0 and Bμ vector boson plane. It is defined as: cosθW = g/√(g2 + g'2) sinθW = g'/√(g2 + g'2) It also gives the relationship between the masses of the W and Z bosons as follows: mZ = mW/cosθW Therefore, Zμ0 = gWμ0/√(g2 + g'2) - g'Bμ/√(g2 + g'2)    = (1/√(g2 + g'2))(gWμ0 - g'Bμ) with mass, mZ = mW/g/√(g2 + g'2) = gf/g/√(g2 + g'2) = f√(g2 + g'2) and γμ = g'Wμ0/√(g2 + g'2) + gBμ/√(g2 + g'2)   = (1/√(g2 + g'2))(g'Wμ0 + gBμ) with mass mγ = 0 It is instructive to look at the degrees of freedom before and after SSB has occurred. At the outset there are 4 degrees of freedom associated with the Higgs field, 2 associated with the massless B field and 6 associated with the 3 massless W fields for a total of 12. After SSB there is a real scalar field H with 1 degree of freedom, 3 massive weak bosons, W± and Z0, with 9, and 1 massless photon with 2, again yielding a total of 12. The scalar field degrees of freedom have been 'eaten' to give the W± and Z0 bosons their longitudinal components. Note: massless particles cannot have a degree of freedom in their longitudinal direction since they are constrained to travel at the speed of light along their direction of motion. Schematically, the before and after SSB scenario is illustrated below: - - - - | H+ + iH- | -> | 0 | | H  + iH0 | | f + H | - - - - Under infinitesinal rotations the vacuum, φ0, should be symmetric. Therefore, φ0 -> exp(iθZ)φ0 where Z is the rotation. This means that, φ0 -> (1 + iθZ)φ0 So Zφ0 must equal 0. SU(2)L: - - - - - - σ3φ0 = | 1 0 || 0 | = -| 0 | ≠ 0 -> broken    | 0 -1 || f + H | | f + H | - - - - - - U(1)Y: - - - - - - YWφ0 = | 1 0 || 0 | = | 0 | ≠ 0 -> broken    | 0 1 || f + H | | f + H | - - - - - - U(1)EM: - - - - Qφ0 = (1/2)(σ3 + YW) = | 1 0 || 0 | = 0 -> unbroken     | 0 0 || f + H | - - - - After SSB, the H+, H- and H0 components all become Goldstone bosons. The 4th component, H, gets a VEV, f. The 3 Goldstone bosons get 'eaten' by combinations of the massless Wμ0, Wμ1, Wμ2 and Bμ field bosons in the following way: The negative combination of Wμ1 and Wμ2 'eats' H+ giving the W+ boson a longitudinal 3rd polarization (i.e. mass). Likewise, the positive combination of Wμ1 and Wμ2 'eats' H- to give the W- boson a mass. And the negative combination of Wμ0 and Bμ 'eats' H0 to give the Z0 boson a mass. Finally, the leftover combination of Wμ0 and Bμ forms the photon. But since there are no more Goldstone bosons to 'eat' the photon remains massless. We could repeat this process with (f + H) to see the involvement of the the Higgs boson. However, the math is more cumbersome and therefore is not attempted here. The end result is the Higgs boson component forms a condensate which, as we will see, can interact with fermions to also give them mass. SU(3) ⊗ SU(2) ⊗ U(1) Symmetry Breaking --------------------------------------- Under a local SU(3) transformation the field transforms as: φ -> φexp(iθi(x)λi/2) where λi are the Gell-Mann matrices - the generators of SU(3). Since there are 8 generators there are 8 gauge fields designated by Gμi which correspond to the gluons. The covariant derivative for a SU(3)⊗SU(2)⊗U(1) gauge invariant Lagrangian is: Dμ = ∂μ + ig''Gμiλi/2 + igWμiσi/2 + ig'YWBμ Where g, g' and g'' are coupling constants. Due to the non-commutativity of the generators of this symmetry, the gauge tensor is: Gμν = ∂μGνi - ∂νGμi + fijkGμjGνk where fijk is the structure constant of the particular gauge group. Fermionic Mass -------------- - - φ = | 0 | e | f + H | - - - - φ = | 0 f + H | e-iα - - The fermions also get their mass from the spontaneous symmetry breaking of the Higgs Field. Fermions include quarks, neutrinos, electrons, muons and tau's. It has been experimentally discovered that in the decay of a neutron to a proton, the electron emitted is always left handed. This turns out to be true for any process involving the weak interaction. For example, an interaction involving a W boson and an electron to produce a neutrino, will only take place if the electron is left handed. In addition, the neutrino produced will always be left handed. For any fermion we can write the Dirac equation in terms of right and left handed particles as follows (h = c = 1): i∂ψR/∂t + iαi∂ψR/∂x = mψL i∂ψL/∂t + iαi∂ψL/∂x = mψR Lets consider the electron. The mass terms combine two particles, eR and eL into a single particle - the 'physical electron' which is propagating through space. eL and eR swap back and forth. At a particular point in time the particle may be eL, but if you observe it a moment later, the very same particle might manifest itself as eR. Weak interactions with the 'physical electron' occur through its left-handed component and not with the right-handed component. At this point we need to consider the weak hypercharge associated with the weak interaction. The weak hypercharge, YW, relates the electric charge, Q, and I3. It satisfies the equality: Q = I3 + YW/2 Where Q is the charge and I3 is the third component of the isospin. ∴ YW = 2(Q - I3) Consider the case for electrons. YW for a left-handed electron, eL, is -1 and for a right-handed electron, eR, is -2. It is clear that, as it stands, plugging these values into the 2 Dirac equations results in a charge violation that 'breaks' the coupling between the two versions of the electron. Now, the Higgs field, H, has a weak hypercharge equal to 1. If we rewrite the Dirac Equations as: i∂ψR/∂t + iαi∂ψR/∂x = mHψL i∂ψL/∂t + iαi∂ψL/∂x = mHψR Under SSB, the weak hypercharge is now conserved. In essence, the Higgs field become a condensate that acts as a source and sink of the weak hypercharge. The addition/subtraction of H (f) can be looked at in the following way: /H (provides weak hypercharge) / rh------gy \ \ lh /H(removes weak hypercharge) / lh------gy \ \ rh Now, H = f (the VEV) and we can write the 2 Dirac equations as (f = f): i∂ψR/∂t + iαi∂ψR/∂x = gyL i∂ψL/∂t - iαi∂ψL/∂x = gyR where gy is the YUKAWA coupling constant and represents the interaction between the Higgs field and a Dirac field. It is apparent from this the mass term in the original equations is equivalent to gyf. Therefore, we can write: m = gyf There is a gy for every type of fermion. Example, for an electron - W boson interaction: gy ~ 90GeV/0.5Mev ~ 10-5 The above calculations have assumed the low energy case where we ignored the oscillations of the Higgs field and assumed that f is a constant. To include the Higgs boson we would need to rewrite the right hand side of the equations as follows: i∂ψR/∂t + iαi∂ψR/∂x = gy(f + H)ψL i∂ψL/∂t + iαi∂ψL/∂x = gy(f + H)ψR Alternative Derivation ---------------------- We can also derive similar results from the Lagrangian of the Dirac equation. The Dirac field has 4 components: Positive and negative energies and left and right handedness (spin). The Dirac equation with h = c = 1 is: (iγμμ - m)ψ = 0 Solutions to this equation are the DIRAC SPINORS. _ _ To get the Lagrangian multiply by ψ where ψ = γ0ψ ... the DIRAC ADJOINT SPINOR. Thus, _ L = ψ(iγμμ - m)ψ _ ψ is interpreted as a creation operator and ψ is interpreted as an annihilation operator. Note: We can prove that this is the Lagrangian by applying the Euler-Lagrange equations. The result yields the original Dirac equation. Also, this is not the Lagrangian for QED. The QED Lagrange contains Dμ and the electromagnetic tensor. - -   - - - - ψ = | ψ1 | ψ = | ψ1 ψ2 ψ3 ψ4 | γ0ψ = | ψ1 ψ234 | | ψ2 |   - - - - | ψ3 | | ψ4 | - - The purpose of γ0 is to make the Lagrangian Lorentz invariant. Rewrite the DE as: _   _ iψγμμψ - mψψ = 0 The first term in the Lagrangian is the kinetic term. The second term flips the chirality back and forth. We can write: The Dirac spinor, ψ, can be written as a linear combinations of left-handed and right-handed states. Therefore, ψ = ψL + ψR and ψ = ψL + ψR _ Thus, using the relationship ψ = ψγ0, we get: _ ψψ = ψγ0ψ - - - - - - = | ψL ψR || 0 1 || ψL | - - | 1 0 || ψR | - - - - = ψLψR + ψRψL The first term of the Lagrangian can be written as: LKE = iψLγμμψL + iψRγμμψR In order to account for the interaction of the fermions with the symmetry group SU(2)⊗U(1) and the gauge fields we have to replace ∂μ with the gauge covariant derivative, Dμ. Thus, LKE = iψLγμDμψL + iψRγμDμψR The second term of the Lagrangian can be written as: _ mψψ = mψLψR + mψRψL However, this as it stands would not be a legal term in the Lagrangian because it violates weak hypercharge conservation. As before, we can restore the coupling if we introduce the Higgs field, H, as follows: _ gyψHψ = gyLR + ψRHψL) _ where gyψHψ is the YUKAWA INTERACTION Consider the case H = f. We get: _   gyψfψ = gyf(ψLψR + ψRψL) Now consider f + H. We get: _   gyψ(f + H)ψ = gyL(f + H)ψR + ψR(f + H)ψL) So the RHS becomes: gyf(ψLψR + ψRψL) + gyH(ψLψR + ψRψL) + the hermitian conjugate The first term is the fermion mass. The second term describes a process whereby a fermion is absorbed and re-emitted alongside a Higgs boson. The Complete Electroweak Lagrangian ----------------------------------- Before SSB: LEW = LHiggs + LGauge + LDirac + LYukawa = DμφDμφ - V(φ) - FμνFμν - GμνiGμνi + LγμDμψL + iψRγμDμψR + gyLφψR + ψRφψL) Majorana Mass ------------- The kind of fermion mass that we discussed above is called a Dirac mass. This is a type of mass that connects two different particles, eR and eL. It is also possible to have a mass that connects two of the same kind of particle, this is called a MAJORANA mass. This type of mass is forbidden for particles that have any type of charge. There is, however, one type of matter particle in the Standard Model which does not carry any charge - the neutrino. Majorana masses mix neutrinos with anti-neutrinos so that the “physical neutrino” is its own antiparticle.